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1 files changed, 29 insertions, 29 deletions
diff --git a/docs/source/physics.rst b/docs/source/physics.rst index ed603f2..4589eea 100644 --- a/docs/source/physics.rst +++ b/docs/source/physics.rst @@ -28,13 +28,13 @@ the neutrino it's name as a play on words of *little neutron* in Italian [4]_. It was not until some 20 years later that the discovery of the neutrino was realised. It was eventually understood that neutrinos came in three distinct -*flavours* :math:`\left (\nu_e,\nu_\mu,\nu_\tau\right )` along with their +*flavors* :math:`\left (\nu_e,\nu_\mu,\nu_\tau\right )` along with their associated antiparticles :math:`\left (\bar{\nu}_e,\bar{\nu}_\mu,\bar{\nu}_\tau\right)`. Neutrino Mixing --------------- -For the three massive neutrinos, the flavour eigenstates of the neutrino +For the three massive neutrinos, the flavor eigenstates of the neutrino :math:`\mid{\nu_\alpha}>`, :math:`\alpha\in\{e,\mu,\tau\}`, are related to the mass eigenstates :math:`\mid{\nu_i}>`, :math:`i\in\{1,2,3\}` via a unitary mixing matrix :math:`U_{\alpha i}` known as the PMNS matrix [5]_, [6]_: @@ -49,12 +49,12 @@ This relationship can be seen better in this image: :width: 500px :align: center - Graphical representation of the relationship between the neutrino flavour and + Graphical representation of the relationship between the neutrino flavor and mass eigenstates. The three mass eigenstates are depicted as three boxes, coloured such that the relative area gives the probability of finding the - corresponding flavour neutrino in that given mass state. + corresponding flavor neutrino in that given mass state. -The time evolution of the flavour eigenstate as the neutrino propagates is +The time evolution of the flavor eigenstate as the neutrino propagates is given by: .. math:: @@ -63,7 +63,7 @@ given by: \sum^3_{i=1}U^*_{\alpha i}\mid{\nu_i\left(t\right)}> The oscillation probability gives the probability that a neutrino produced in a -flavour state :math:`\alpha` is then detected in a flavour state :math:`\beta` +flavor state :math:`\alpha` is then detected in a flavor state :math:`\beta` after a propagation distance :math:`L`: .. math:: @@ -98,7 +98,7 @@ where :math:`\Delta m_{ij}^2=m_i^2-m_j^2`. Note that for neutrino oscillations to occur, there must be at least one non-zero :math:`\Delta m_{ij}^2` and therefore there must exist at least one non-zero neutrino mass state. -The mixing matrix can be parameterised using the standard factorisation [8]_: +The mixing matrix can be parameterized using the standard factorization [8]_: .. math:: @@ -126,7 +126,7 @@ where :math:`s_{ij}\equiv\sin\theta_{ij}`, :math:`c_{ij}\equiv\cos\theta_{ij}`, violating phase. Overall phases in the mixing matrix do not affect neutrino oscillations, which only depend on quartic products, and so they have been omitted. Therefore, this gives a total of six independent free parameters -describing neutrino oscillations for three neutrino flavours in a vacuum. This +describing neutrino oscillations for three neutrino flavors in a vacuum. This table outlines the current knowledge of these parameters determined by a fit to global data [9]_: @@ -134,7 +134,7 @@ global data [9]_: :width: 500px :align: center - Three neutrino flavour oscillation parameters from a fit to global data + Three neutrino flavor oscillation parameters from a fit to global data [9]_. This table shows two columns of values, *normal ordering* and *inverted @@ -205,9 +205,9 @@ coincidence with neutrinos coming from a particular source has successfully been able to identify for the very first time, a source of high-energy astrophysical neutrinos [11]_, [12]_. -Of particular interest is the composition of flavours produced at the source. -In the simple pion decay model described above, the *neutrino flavour -composition* (sometimes referred to as the *neutrino flavour ratio*) +Of particular interest is the composition of flavors produced at the source. +In the simple pion decay model described above, the *neutrino flavor +composition* (sometimes referred to as the *neutrino flavor ratio*) produced at the source is: .. math:: @@ -215,12 +215,12 @@ produced at the source is: \pi\text{ decay}\rightarrow \left(f_e:f_\mu:f_\tau\right)_\text{S}=\left(1:2:0\right)_\text{S} -For all discussions on the astrophysical neutrino flavour composition, the +For all discussions on the astrophysical neutrino flavor composition, the neutrino and antineutrino fluxes will been summed over as it is not yet experimentally possible to distinguish between the two. In the case that the muon interacts in the source before it has a chance to decay, e.g.\@ losing energy rapidly in strong magnetic fields or being absorbed in matter, only the -:math:`\nu_\mu` from the initial pion decay escapes and so the source flavour +:math:`\nu_\mu` from the initial pion decay escapes and so the source flavor composition is simply: .. math:: @@ -239,19 +239,19 @@ decay, :math:`n\rightarrow p+e^-+\bar{\nu}_e`, which gives rise to a purely Production of :math:`\nu_\tau` at the source is not expected in standard astrophysics models. However, even in the standard construction, the composition could vary between any of the three idealised models above, which -can be represented as a source flavour composition of :math:`(x:1-x:0)`, where +can be represented as a source flavor composition of :math:`(x:1-x:0)`, where :math:`x` is the fraction of :math:`\nu_e` and can vary between :math:`0\rightarrow1`. Once the neutrinos escape the source, they are free to propagate in the vacuum. -As discussed above, neutrinos can transform from one flavour to another. +As discussed above, neutrinos can transform from one flavor to another. Astrophysical neutrinos have :math:`\mathcal{O}(\text{Mpc})` or higher baselines, large enough that the mass eigenstates completely decouple. The astrophysical neutrinos detected on Earth are decoherent and are propagating in pure mass eigenstates. Taking this assumption greatly simplifies the transition probability as all the interference terms between the three mass eigenstates can be dropped, and all that is left is to convert from the propagating mass -state to the flavour states: +state to the flavor states: .. math:: @@ -259,13 +259,13 @@ state to the flavour states: \phi_{\alpha,\oplus}&=\sum_{i,\beta} \mid{U_{\alpha i}}\mid^2\mid{U_{\beta i}}\mid^2\phi_{\beta,\text{S}} -where :math:`\phi_\alpha` is the flux for a neutrino flavour :math:`\nu_\alpha` +where :math:`\phi_\alpha` is the flux for a neutrino flavor :math:`\nu_\alpha` and :math:`\phi_i` is the flux for a neutrino mass state :math:`\nu_i`. The subscript :math:`\text{S}` denotes the source and :math:`\oplus` denotes as measured on Earth. The same result can be obtained in the plane wave picture of the neutrino mixing equations above and taking the limit :math:`L\rightarrow\infty`, thus this type of decoherent mixing is also known -as oscillation-averaged neutrino mixing. From this, the flavour composition on +as oscillation-averaged neutrino mixing. From this, the flavor composition on Earth is defined as :math:`f_{\alpha,\oplus}=\phi_{\alpha,\oplus}/\sum_\alpha\phi_{\alpha,\oplus}` and this can be calculated using the mixing matrix parameters the table above. @@ -281,15 +281,15 @@ For the three source models discussed above: This can be visualised in a ternary plot, which you can make yourself by checking out the :doc:`examples` section! The axes here are the fraction of -each neutrino flavour as shown below. The coloured circle, square and triangle -show the source flavour compositions. The arrows show the effect of neutrino -mixing on the flavour composition. The unfilled circle, square and triangle -show the corresponding measured flavour composition. Neutrino mixing during -propagation has the effect of averaging out the flavour contributions, which is +each neutrino flavor as shown below. The coloured circle, square and triangle +show the source flavor compositions. The arrows show the effect of neutrino +mixing on the flavor composition. The unfilled circle, square and triangle +show the corresponding measured flavor composition. Neutrino mixing during +propagation has the effect of averaging out the flavor contributions, which is why the arrows point towards the centre of the triangle. This effect is more pronounced for :math:`\nu_\mu\leftrightarrow\nu_\tau` due to the their larger mixings. Also shown on this figure in the hatched *Standard Model* area, is the -region of measured flavour compositions containing all source models of +region of measured flavor compositions containing all source models of :math:`\left(x:1-x:0\right)`, using Gaussian priors on the standard mixing angles. Therefore, this hatched area is the region in which all standard astrophysical models live. @@ -298,11 +298,11 @@ astrophysical models live. :width: 700px :align: center - Astrophysical neutrino flavour composition ternary plot. Axes show the - fraction of each neutrino flavour. Coloured shapes show 3 models for the - source flavour composition. The arrows indicate the effect of neutrino mixing + Astrophysical neutrino flavor composition ternary plot. Axes show the + fraction of each neutrino flavor. Coloured shapes show 3 models for the + source flavor composition. The arrows indicate the effect of neutrino mixing during propagation and the unfilled shapes show the corresponding measured - flavour compositions. The hatched area shows the region in measured flavour + flavor compositions. The hatched area shows the region in measured flavor space in which all standard astrophysical models live. IceCube |